Department of Mathematics
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Item Metric Dimension Of Different Chemical Structures(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Muhammad Umer Farooq; SP17-RMT-001; Dr. Muhammad Hussain; LHR TP 5431Item Stationary Solutions And Optical Solitons For Nonl(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Insibat Afzal; FA17-RMT -012; Dr. Syed Tahir Raza Rizvi; LHR TP 5649Solitons are vastly growing field of research not only in the field of nonlinear optics but they also appear in applied mathematics, telecommunication industry and in various branches of physics and biology. Specially, optical solitons arises in many different f ields including nonlinear optics. They are the light pulses that do not diffract in the mediumthrough which they passes. These days, optical solitons play a significant role in many branches of modern telecommunication mediums including internet industry, twitter, Facebook and many more. Due to the property of not changing shape and path after collision, optical solitons are used to transfer data without using repeaters. A lot of work have been done in retrieving optical and exact solutions of the solitary waves in nonlinear optics. In this dissertation, we will obtain the stationary solutions of Gerdjikov-Ivanov (GI) equation, cubic-quintic NLSE and paraxial equation by using the help of Lie group analysis. And also, we will get chirped sub-pico bright, dark and dark singular combo optical soliton solutions for Triki-Biswas equation by using some constraints. To solve Triki-Biswas equation we will use two different integrating methods namely, sine Gordon method and modified extended direct algebraic method.Item Space Spectral Time Fractional Finite Difference Method along with Stability Analysis for Fractional Order Nonlinear Wave Equations(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Muhammad Sarmad Arshad; SP13-PMATH-004; Dr. Ayesha Sohail; LHR TP 5299In this work, nonlinear partial differential equations governing the obscure phenomena of shallow water waves are discussed. Time fractional model is considered to understand the upcoming solutions on the basis of all historical states of the solution. A semi-analytic technique, Homotopy Perturbation Transform Method (HPTM) is used in conjunction with a numerical technique to validate the approximate solutions. With the aid of graphical interpretation, the favorable wave parameters, to avoid wave breaking are estimated. Afterwards, dynamical analysis of fractional order Schr dinger equation governing the optical wave propagation is reported in detail. The validity criteria for the application of the semi-analytic asymptotic methods are exploited. Comparison between the solutions obtained by the two asymptotic techniques, that is, the Fractional Homotopy Analysis Transform Method and the Optimal Homotopy Analysis Method is performed to select the most accurate technique for the stated problem. Space spectral analysis with integrating factor technique and time fraction finite difference method have been implemented to study the pressure waves propagating in bubbly fluids as well as nonlinear phenomena of plasma waves. Dynamical analysis of acoustic/pressure waves propagating in bubbly fluids is of great significance. Such flows arise in many engineering problems including sonochemistry, sonochemical reactors, cavitation around hydrofoils and ultrasonic propagation in medicine and biology. Fractional approach for modeling the propagation of the pressure waves in liquids containing a large number of tiny gas bubbles is proposed. Moreover, numerical solution of the fractional order Modified Korteweg-de Vries equation governing the dynamics is approximated using a novel space spectral time fractional finite difference tool. A spectral technique for space and a multi-step finite difference scheme for time are designed and implemented. The spatial spectral discretization error and the stability bounds are discussed. The nonlinear phenomena of plasma waves are well demonstrated with the aid of graphical analysis. Stability analysis of integer and fractional order KdV equations have been discussed quantitatively with the help of Evans function approximation.Item Different Cases of Singular Domains in Two Population Synaptic Drive Model(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Wasim Shahid; SP16-RMT-008; Dr. Muhammad Yousaf; LHR TP 5198In this work we have investigated the dynamics of synaptic drive two-population neural firing rate model for the case of steep firing rate function by applying the theory of switching dynamical systems. The switching dynamic theory simplifies the dynamical system and receive the approximation of the solution for the system with certain modes of parameters. In such type of outlook, the phase space is divided into regular as well as singular domains. These singular domains are characterized into black, white and transparent on the basis of the behavior of trajectories on the sides of singular domain. Moreover, there has been investigation of the existence and uniqueness solutions with corresponding singular and regular stationary points. It has been observed that regular stationary point, which are located within regular domain, are always constant and this characteristic is secured for smooth and sufficiently steep firing rate function. Yousaf.et al investigated the dynamics for the same model. They investigated the dynamics only for some particular cases. We have explored some more cases of singular domains and investigated the dynamics in regular and singular domains. Moreover, general formulae for the stationary points of the system have been calculated and the existence of singular stationary points which belong to the singular domains have been investigated. Furthermore, these singular domains in new dynamical system have also been transformed. Besides, the stability of singular stationary points has been investigated. In the wake of outcomes, the stationary points are attended by implementing the singular perturbation analysis and Thikhonov theorem.Item T-Norms Based Fuzzy Ordering And Their Application(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Atiq-Ur-Rehman; SP12-PMATH-001; Dr Samina Mazhar; LHR TP 4874This research work focuses on T-transitive fuzzy orderings, their mathematical structures and representation results on these orderings. We have shown that the consistent nature of a fuzzy preference relation has direct relationship with its being more or less T-transitive. After handling this task, some work is done on applications of T-transitivity in the area of preference modeling. Here we have presented some new algorithms to complete an incomplete fuzzy preference relation. These algorithms are based on T-transitivity of a given fuzzy relation. This work has its roots in multi criteria and multi agent decision making. The target is to achieve a ranking among alternatives while incomplete information is provided by the judges about criteria of selection and pair wise preferences amongst alternatives. We have further extended these results to their interval valued versions where the consistent and consensus degrees are also accommodated. The theory and algorithms provided so far are then used to solve real life problems of industry i.e., the problem of supplier selection in the supply chain management where the successful implementation of the results is demonstrated.Item Imex Methods For Ordinary Differential Equations(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Nida Khalid; FA17-RMT-053; Dr. Yousaf Habib; LHR TP 5674The aim of this thesis is to construct partitioned Runge-Kutta methods (PRK) for the numerical solution of non-separable system of ordinary differential equations y0 = f(y; z), z0 = g(y; z) as IMEX scheme (implicit explicit). The partitioned Runge-Kutta methods consist of two RK methods such that one method solve y0 = f(y; z), the non stiff differential equation, the other method solve z0 = g(y; z), the stiff differential equation. Explicit RK method is used for the solution of non stiff part and implicit RK method is used for the solution of stiff part. In order to construct PRK method, we have used the idea of effective order methods. The development of these methods require the solution of large number of order conditions, however, there exist simplifying assumptions which reduce the number of order conditions. In this thesis, simplifying assumptions for PRK method for non-separable differential equations are derived for the first time and applied successfully to reduce the effective order conditions, thus allowing us to construct effective order 3 PRK method with just two stages. This result in huge savings in terms of computational cost.Item Particle Creation with Adiabatic Transformation Through Thermodynamic Perspective in Modified Theories(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Mubashira Rasheed; FA17-RMT -014; Dr. Muhammad Zubair; LHR TP 5651Modified gravitational theories play significant role to discuss many con stituent of the accelerated phase of our universe for last few decades. A large numberofdifferent techniques in modified theories are proposed to sort out the cosmological problem. Weanalyzedopenirreversiblethermodynamicsinthescenarioofparticle cre ation for modified theory of gravity known as f(G,T) gravity theory, in which the generic function defines the coupling between GB invariant G and trace of the EMT with non-zero divergence. We discussed the generalized conser vation equation which involves additional contribution from matter geome try coupling for irreversible matter creation process with the help of general ized "second law of thermodynamics" in the light of the “Friedmann-Lematre Robertson-Walker" FLRW space time. Particle creation rate, number of particle creation, creation pressure, entropy production rate, and temperature for this theory have been formulated by choosing some particular model of f(G,T), which help us to explore the particle creation in a significant way. We also discussed non-minimal F(R,T,Q) theory of gravity where (R, T, Q are ricci scalar, trace of EMT and interaction of Ricci tensor and non-minimal coupling of Ricci scalar respectively, and Q = RαβTαβ) is the interaction of Ricci tensor and EMT. Due to the non-minimal coupling of gravity and geometry an extra force present which is perpendicular to four velocity. We also formulated some explicit results of created particle from gravitational field to the created ix matter such as particle creation rate, pressure of created particle and entropy with respect to time variation through irreversible thermodynamics interrelated with adiabatic transformation in the context of FLRW space time. We consid ered the dust universe i.e., P = 0 and the scale factor by power law a(t) = tm to precise our results. It is observed that in this procedure a large amount of co-moving entropy has been generatedItem Randic, Seidel And Laplacian Energy Of NEPS Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Yasir Ali; FA17-RMT-007; Dr. Sarfarz Ahmad; LHR TP 5645Let be the simple graph, then we can obtained the energy of a graph by taking the absolute sum of the eigenvalues of the adjacency matrix of . In this research we have computed different energy invariants of the Non-completed Extended P-Sum (NEPS) of graph . In particular we investigate the Randic, Seidel and Laplacian energies of the NEPS of path graph with any base . Here denotes number of vertices and denotes the number of copies of path graph . Our some of the results depend on the number of zeroes in base elements, for which we use the notation .Item Metric Dimension Of Different Chemical Structures(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Hafiz Muhammad Bilal; FA16-RMT-027; Dr. Muhammad Hussain; LHR TP 5191The Metric Dimension of a graph G is the minimum number of basis element in a resolving set. Let G = (V,E) be a connected graph and length of a shortest path between u and v is known as distance, denoted by d(u,v) in G. Let B = {b1,b2,...,bk} be an ordered set of vertices of G. The representation r(u|B) of u with respect to B is the k − tuple {d(u,b1),d(u,b2),d(u,b3),...,d(u,bk)}, where B is called a resolving set or locating set if every vertex of G is uniquely identified by its distances from the vertices of B or equivalently if distinct vertices of G have distinct representations with respect to B. A resolving set of minimum cardinality is called a basis for G and cardinality is the metric dimension of G is denoted by dim(G). In this thesis we investigated metric dimension of subdivided honeycomb network and Aztec diamond networkItem Consensus Based Group Decision Making With Incomplete Reciprocal Fuzzy Preference Relations Using Product T-norm.(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Mustanser Hussain; FA17-RMT-035; Dr. Adeel Farooq; LHR TP 5661In this dissertation, a consensus-based method for group decision making (GDM) using product transitivity with incomplete fuzzy preference relations (IFPRs) is proposed. Additionally, an average aggregation operator has been used at the first level to estimate the missing preference values and construct the complete fuzzy preference relation (FPR). Then it is confirmed to be product consistent by using the transitive closure formula. Following this, weights of decision makers (DMs) are evaluated by merging consistency weights and predefined priority weights (if any). The consistency weights for the DMs are estimated through product consistency investigation of the information provided by each DM. The consensus process determines whether the selection procedure should be initiated or not. The hybrid comprises of a quitting process and feedback mechanism, and is used to enhance the consensus level amongst DMs in case of an inadequate state. The quitting process arises when some DMs decided to leave the course, and is common in GDM while dealing with a large number of alternatives. The feedback mechanism is the main novelty of the proposed technique which helps the DMs to improve their given preferences based on this consistency. At the end, a numerical example is deliberated to measure the efficiency and applicability of the proposed method after the comparison with some existing models under the same assumptions. The results show that proposed method can offer useful comprehension into the GDM process. ix
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